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User blog:P進大好きbot/Guideline on How to Use Large Cardinals for Ordinal Notations
This is a simle guideline on how to use large cardinal axioms to construct an ordinal notation in \(\textrm{ZFC}\) set theory. # Choose any large cardinal(s) \(K_{\textrm{big}}\). # Define an ordinal collapsing function(s) \(\Psi_{\textrm{great}}\) under \(\textrm{ZFC} + K_{\textrm{big}}\). # Define an ordinal notation, i.e. a countable set \(C\) equipped with a map \(o \colon C \to \textrm{ON}\), by translating constants and functions into constant term symbols and function symbols under \(\textrm{ZFC} + K_{\textrm{big}}\). # Compute the binary relation \(\alpha <_C \beta \stackrel{\textrm{def}}{\Leftrightarrow} o(\alpha) \in o(\beta)\) under \(\textrm{ZFC} + K_{\textrm{big}}\). # Analyse the ordinal type of segments of \((C,<_C)\) by comparing other known countable ordinals, e.g. \(\textrm{PTO}(A_{\textrm{strong}})\) for some an arithmetic or a set theory \(A_{\textrm{strong}}\), under \(\textrm{ZFC} + K_{\textrm{big}}\). # Prove that \(<_C\) admits a (recursive) definition under \(\textrm{ZFC}\). # Prove that \(<_C\) is well-founded under \(\textrm{ZFC}\). # Prove that the analysis on \((C,<_C)\) can be reduced to \(\textrm{ZFC}\). None would forget to check Step 1, 2, 3, 4, and 5, but I sometimes saw googologists to forget to check Step 6, 7, and 8. I explain more about it. = Definition under ZFC = If there is no interpretation of \(<_C\) into a definition under \(\textrm{ZFC}\), then \((C,<_C)\) is useless for googologists to construct a well-defined large number under \(\textrm{ZFC}\). Moreover, if there is no recursive way to compute a large number under \(\textrm{ZFC}\), then the resulting large number becomes uncomputable. For example, one might need recursive definition of \(<_C\), the subset of standard form, and the system of the fundamental sequence, and so onThere are several ways to use non-recursive ordinal notations to define a computable large number, and hence the recursiveness is not necessarily needed.. The definition of \((C,<_C)\) under \(\textrm{ZFC} + K_{\textrm{big}}\) just ensures that the existence of such a \((C,<_C)\) is consitent with \(\textrm{ZFC}\) under the assumption of \(\textrm{Con}(\textrm{ZFC} + K_{\textrm{big}})\). It does not imply that \((C,<_C)\) is also well-defined under \(\textrm{ZFC}\). = Well-foundedness under ZFC = The binary relation \(<_C\) is well-founded by the definition under \(\textrm{ZFC} + K_{\textrm{big}}\), because \((\textrm{ON},\in)\) is well-founded. However, the well-foundedness of \(<_C\) under \(\textrm{ZFC} + K_{\textrm{big}}\) just ensures that the well-foundedness of \(<_C\) is consistent with \(\textrm{ZFC}\) under the assumption of \(\textrm{Con}(\textrm{ZFC} + K_{\textrm{big}})\). It does not imply that the ordinal type of \((C,<_C)\) is well-defined under \(\textrm{ZFC}\). It would be no longer called as an ordinal notation, but just a notationIf one assumes the existence of an \(\omega\)-model of \(\textrm{ZFC} + K_{\textrm{big}}\), then the termination of the computation of the computable function \(f_{\alpha}(n)\) by recursion on \(<_C\) might be provable pointwise on each standard natural number \(n\) and a term \(\alpha \in C\) presented by a formal string of standard natural numbers, because the value of \(f(n)\) merely depends on a model. However, it does not imply that \(f_{\alpha}(n)\) is provably total. Therefore you can only argue the growth rate restricted to standard natural numbers, which is not so useful to analyse other large functions, because the growth rate of a total computable function is not estimated by that of a partial function.. It is possible that a recursive analogue \(K_{\textrm{big}}^{\textrm{CK}}\) of \(K_{\textrm{big}}\) would work well, as long as you have a proof of the statement that the ordinal notation associated to \(K_{\textrm{big}}^{\textrm{CK}}\) is isomorphic to \((C,<_C)\). Otherwise, the recursive analogue does not work as you expect. The word "recursive analogue" is not an almighty magic spell. It works only when you have proofs of non-trivial results. The existence of an isomorphism under \(\textrm{ZFC} + K_{\textrm{big}}\) does not ensure the isomorphism under \(\textrm{ZFC}\). Something observation like "the collapsing process looks similar" never implies the isomorphism under \(\textrm{ZFC}\), either. = Analysis under ZFC = The analysis of \((C,<_C)\) heavily depends on \(K_{\textrm{big}}\). A result like \(\Psi_{\textrm{great}}(\varepsilon_{K_{\textrm{big}}+1}) = \textrm{PTO}(A_{\textrm{strong}})\) under \(\textrm{ZFC} + K_{\textrm{big}}\) just ensures that the statement that the limit of \((C,<_C)\) is beyond \(\textrm{ZFC} + K_{\textrm{big}}\) is consistent with \(\textrm{ZFC}\) under the assumption of \(\textrm{Con}(\textrm{ZFC} + K_{\textrm{big}})\). It does not imply that \((C,<_C)\) has that great strength under \(\textrm{ZFC}\). If you want to analyse \((C,<_C)\) under \(\textrm{ZFC}\), then you need to refer to other known results under \(\textrm{ZFC}\), or sincerely verify your desired "fact" by studying the proof-theoretic ordinal analysis, the reflection property, or something like that. = Footnotes = Category:Blog posts